Bounded λ-Connectedness for Three-Dimensional Segmentation: Seeds, Boundary Merging and Public Geophysical Data

We extend λ-connected segmentation from two to three dimensionswith the aim of a practical method for volumes whose objects varygradually, such as seismic velocity models and medical CT. The guid-ing rule is to use the simplest method that works: a band threshold,or pure λ-connectedness when the object has a good property, and acombined method only when it fails. We study a bounded form ofλ-connectedness in which a region must be λ-chain connected and itsmaxmin range must not exceed a bound B . Pure λ-connectednessis the case where B is the whole data range; a fixed window gives anequivalence relation; an adaptive bound gives only a similarity relation.For the case where one does not know what to segment, we describe apipeline of λ-core seeds, a marker watershed on the largest neighbourstep, and a boundary merge in which two regions join when most oftheir border voxel pairs satisfy λ. Synthetic volumes with known truthshow where single-link merging leaks, where all-pairs merging is fooledby one outlier, and that the most-pairs rule is independent of regionlabelling. On a public USGS 3-D shear-velocity model (CC0) we useno geological labels: units are dened by velocity. A deep block is seg-mented into four regions that stay stable when the lateral resolution isreduced by factors of two to eight (adjusted Rand index 0.936-0.984);a shallow block shows a steep depth ramp, which the bounded methodcuts where unbounded growth did not; an intermediate-depth blockis a smooth field in which one region is the correct answer; and adepth-detrended band threshold, with the bound chosen by k -meansrather than Otsu, delineates one connected low-velocity body in thetop 2.4 km. We state plainly what is not yet shown: no ground truthexists for the real volumes, and several parameters were chosen byinspection.